STRUCTURE AND INTERPRETATION OF MUSIC CONCEPTS
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    CHAPTER-3 -- TEMPORAL STRUCTURES, TEMPORAL   PROCESSES, RHYTHM CONCEPTS
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                             Homework no. 3
    
Due: Jan 1, 03

Please demonstrate every procedure that you construct.

1. The temporal structure constructors already implemented in
   chapter Piece-process-3 use "lazy constructors". That is, the constructors
   just combine the component temporal-structures and their associated
   time-intervals into a new composite value.
   Alternatively, we could implement "eager constructors": constructors that
   optimize the constructed value and compute its properties. For example:
      * The composite TS constructor (cts) could sort the components either
        by the scheduling time-intervals (the ti-s), or by the ti +
        duration(ts).
      * The constructor can shift all time-intervals so that at least one
        time-interval is (time 0) (there is no "vacuum" between the (time 0)
        of the composite temporal-structure and the time point where the first
        temporal-structure is scheduled).
      * Moreover, a constructed TS could compute once and for all its duration
        and its process.
   Implement eager constructors for the Composite-TS and the Relative-TS
   temporal structures. Use these constructors for optimized implementations for
   the simultaneous and the sequential temporal structures.

   Demonstrate the advantages of the optimized implementation.

2. The Process implementation in chapter Piece-process-3 imitates the
   hierarchical construction of temporal structures. That is, the process
   associated with a composite temporal structure of some kind is constructed
   from the processes of its components.
   Instead, we could have only a single kind of "flat" process: A time
   procedure with multiple scheduled play-notes:
       (lambda(tp)(<a loop that applies play-note to all notes in a flat TS>)
   For that purpose we need a procedure flatten-cts (flatten-rel-cts) that
   flattens a composite-temporal-structure (or rel-cts) into a cts (or rel-cts)
   whose components are atomic (play-note can be applied to them, either
   directly or following a ->note transformation.

   Discuss: Is there a need for hierarchical processes?
   
3. Use the Composite-TS (or Rel-TS) constructors for an n-depth round
   temporal-structure. That is:

   depth 0: ts
   depth 1: cts(ts ti ts)
   depth 2: cts( cts(ts ti ts) 2ti cts(ts ti ts))
   depth 3: cts( cts( cts(ts ti ts) 2ti cts(ts ti ts))
                 4ti
                 cts( cts(ts ti ts) 2ti cts(ts ti ts)))
   ...
   depth n: cts( <depth n-1> 2^(n-1)*ti <depth n-1> )

4. RHYTHM (section 3.3).
   A Rhythm is a temporal pattern associated with a rhythm tree, whose front is
   admitted by the temporal pattern.
   A METERED-RHYTHM is a rhythm associated with a meter (a
   relative-time-interval). The depth-1 nodes in the tree are labeled by the
   meter (each depth-1 node stands for a measure).
   The meter imposes a beat periodic structure on the rhythm. Standard meters
   have known beat patterns, with a scale of beat emphasis. There are strong
   beats, weak beats, weakest beats. They are described in the notes given
   by Dalia.

   a. The combination of:
          * A metered-rhythm.
          * A metronom.
          * A sequence of temporal-structures.
      can yield a new temporal-structure:
      1. The metered-rhythm + metronom yields a sequence of (real)
         time-intervals. The sequence should be associated with
         measure-beat-based access tools:
          * Find the time-point occurring on beat i in measure j.
          * Find the sequence-element occurring on beat i in measure j (if any).
          * Find the measure-beat of a sequence element.
      2. The Relative-TS type can be provided with another constructor, that
         takes a sequence of time-intervals and a sequence of
         temporal-structures as parameters.
      Note: The time-interval sequence that results from the metered-rhythm
            does not include the (time 0) onset of the first temporal-structure
            in the sequence.

      Construct the metered-rhythm->rel-cts constructor, and the associated
      tools.
      
   b. A metered-rhythm is classified as stable/unstable if time-intervals that
      fall on strong beats are longer than others.
      Construct a metered-rhythm constructor that takes a stable/unstable
      characterization and constructs an appropriate rhythm.

   c. A metered-rhythm can be annotated with the intended speed
      characterization: fast (vivace), slow (andante), very-slow (largo), etc.
      The speed characterization implies certain metronom values that must be
      used in turning the rhythm into a (real) temporal structure.
      Extend the metered-rhythm->rel-cts constructor from (a) to take an
      'attributed-metered-rhythm' as a parameter. If the metered-rhythm has a
      speed attribute, every realization of the rhythm into a time structures
      restricts the metronom according to the speed characterization.